Entanglement witnesses arising from exposed positive linear maps

TL;DR

This paper introduces a minimal set of entanglement witnesses derived from exposed positive linear maps, capable of detecting all entanglement, including PPT states, with new explicit examples in three-dimensional matrix algebra.

Contribution

It demonstrates that entanglement witnesses from exposed extremal positive maps form a minimal, comprehensive set for entanglement detection and provides the first explicit examples of such maps in 3x3 matrices.

Findings

Every entanglement can be detected by these witnesses.

They detect a unique set of entanglement.

Provided a family of indecomposable maps in 3x3 matrices.

Abstract

We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entanglement can be detected by one of these witnesses, and this witness detects a unique set of entanglement among those. Therefore, they provide a minimal set of witnesses to detect all entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entanglement with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps between three dimensional matrix algebra.